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both the University and the Academy, and was also appointed
professor at the Berlin Wax College.
Kumrner was one of those rarest of all scientific geniuses who
are first class hi the most abstract mathematics, the applica-
tions of mathematics to practical affairs, including war, which
is the most unblushingly practical of all human idiocies, and
finally in the ability to do experimental physics of a high degree
of excellence. His finest work was in the theory of numbers
where bis profound originality led him to inventions of the very
first order of importance, but in other fields - analysis, geome-
try, and applied physics - he also did outstanding work.
Although Kummer's advance in the higher arithmetic was of
the pioneering sort that justifies comparing him with the
creators of non-Euclidean geometry, we somehow get the
impression, on reviewing his life of eighty-three years, that
splendid as his achievement was, he did not accomplish all that
he must have had in him. Possibly his lack of personal ambition
^an instance is given presently), his easy-going geniality, and
his broad sense of humour prevented him from winding himself
hi an attempt to beat the record.
The nature of what Kummer did in the theory of numbers
has been described in the chapter on Rronecker: he restored tiit
fundamental theorem of arithmetic to those algebraic number fields
ixMch arise in the attempt to prove Fermafs Last Theorem and in
the Gaussian theory of cyclotomy, and he effected this restoration
by the creation of an entirely new species of numbers, his so-catted
""ideal numbers'. He also carried on the work of Gauss on the law
of biquadratic reciprocity and sought the laws of reciprocity for
degrees higher than the fourth.
As has already been mentioned in preceding chapters, Rum-
mer's 'ideal numbers' are now largely displaced by Dedekind's
^deals', which will be described when we come to them, so it is
not necessary to attempt here the almost impossible feat of
explaining in untechnical language what Kummer's 'numbers*
are. But what he accomplished by means of them can be stated
with sufficient accuracy for an account like the present: Kum-
mer proved that aP + y^ = zp, where p is a prime, is impossible
in integers x%& all different from, zero, for a whole very exten-